# [seqfan] Uphi(m)=UPhi(n)=3/4*(m*n)^(1/2)

koh zbi74583.boat at orange.zero.jp
Tue Nov 4 07:20:21 CET 2008

```    Hi, Seqfans
Definition of Amicable Number is the following.

Sigma(x) = Sigma(y) = x+y

If "Sigma" is replaced with the other divisor functions or "x+y" is replaced with high degree or rational or irrational formulas then it becomes generalized Amicable Number.

I calculated the examples of the following equtation.

UnitaryPhi(x) = UnitaryPhi(y) = 3/4*(x*y)^(1/2) , y<=x

x=y=2^2
x=y=2^3*7
x=y=2^4*5
x=y=2^5*5*31
x=y=2^6*3^2*7
x=y=2^6*3^3*7*13
x=y=2^7*3^2*7*127
x=y=2^7*3^3*7*13*127
x=y=2^8*5*17
x=y=2^9*3^2*7*73
x=y=2^9*3^3*7*13*73
x=y=2^10*3^3*5^2*11*13*31

x=5*3^2*11^2,      y=5*31^2
x=2^11*5*3^2*89^2, y=2^11*5*11^2*23^2

Could anyone do exhausitive search?
I would like to know the other examples of the case not{x=y}.

%I A000001
%S A000001 4, 56, 80, 4032, 4960, 5445, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 729999360
%N A000001 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m
%C A000001 a(6) and a(13) are the cases not{n=m}
%e A000001 Factorization of cases not{n=m}
m,n=5*3^2*11^2,5*31^2
m.n=2^11*5*3^2*89^2,2^11*5*11^2*23^2
%Y A000001 A000002
%K A000001 none
%O A000001 0,1
%A A000001 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp

%I A000002
%S A000002 4, 56, 80, 4032, 4960, 4805, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 655452160

%N A000002 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m
%C A000002 a(6) and a(13) are the cases not{n=m}
%Y A000002 A000001
%K A000002 none
%O A000002 0,1
%A A000002 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp

Yasutoshi

```